Dually degenerate varieties and the generalization of a theorem of Griffiths--Harris

Abstract

The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a variety X \subset P^N with a degenerate Gauss map of rank r. For such a variety, in the general case, the dimension of its dual variety X* is N - l - 1, where l = n - r, and X is dually degenerate if dim X* < N - l - 1. In 1979 Griffiths and Harris proved that a smooth variety X \subset P^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety X \subset P^N with a degenerate Gauss map of rank r.